Tagged Questions

8
votes
4answers
178 views

How to find $\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$?

Integration by parts is of no success. What else to try? $$\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$$
4
votes
1answer
74 views

An integral involving the error function

I have in my notes the following problem. I recall it being quite difficult and needing a change of variables into polar or spherical coordinates. Assuming I have not made a typo, there is a nice ...
6
votes
2answers
112 views

Trying to recall an integration trick

In my notes, I have the following problem. Find the volume of (a) $x^2+y^2 \le 1$, $x^2+z^2\le 1$ in $\mathbb R^3$ (b) $x^2+y^2 \le 1$, $x^2+z^2\le 1$, $y^2+z^2\le 1$ in $\mathbb R^3$ ...
13
votes
2answers
237 views

Evaluate Integral (Romanian Olympiad)

$$ \int\cos x\cdot\cos^2(2x)\cdot\cos^3(3x)\cdot\cos^4(4x)\cdot\ldots\cdot\cos^{2002}(2002x)dx $$ Taken from the 2002 Romanian olympiad
3
votes
4answers
161 views

Integrals from MIT integration bee

$\int\frac{dx}{2+2\sin x+\cos x}$ $\int_0^{\infty}\frac{\ln x}{1+x^2}dx$ $\int\frac{dx}{x(1+x^3)}$ In general what is $\int \frac{dx}{a+b\sin x}$?
1
vote
1answer
122 views

how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...
0
votes
2answers
131 views

Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions

Let $S=\{f:\mathbb{R} \to \mathbb{R}\}$ that satisfies: $\forall f\in S$, $f'$ exists and $f'$ is continuous and $f(0)=f(1)=0$. Please prove that $\exists \delta :\forall f\in S$ s.t. $\int_{0}^{1} ...
4
votes
0answers
51 views

Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.

Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
16
votes
3answers
368 views

Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.

Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history. Here's a final-round calculus ...
16
votes
2answers
1k views

Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function $f : [0,1] \to ...
4
votes
1answer
858 views

1981 Putnam problem A-3 : Differentiating under a double integral?

I was trying some old problems and got stuck on this one. Then when I looked at the answer there was a step I could not understand. Perhaps you can explain it to me. A-3 Find $ \displaystyle ...